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Discrete Time Multivariable
daptive ontrol
GRAHAM C. GOODWIN, PETER J. RAMADGE, AND PETER E. CAINES
ADAPTIVE
control, so great is its appeal, has been stud
ied for almost forty years. Its early history is one of diverse,
but interesting, heuristic endeavors. One of the first problems
studied, and one that has formed the focus of much subsequent
research, is that ofadaptivemodel reference control inwhich the
controller employs an estimate of the unknown plant to cause
the output y to track y*
= H r,
where
r
is the reference,
H
a
prespecified reference model, and y* the desired output. An
other consistent theme is certaintyequivalence, regarding (and
using) the estimated plant as if it were the true plant; the moti
vation is obvious . . . if the estimated parameters converge to the
true parameters, the certainty equivalent controller converges to
a satisfactory (optimal) controller for the true plant. However,
adaptive controllers can perform satisfactorily even if, as is of
ten the case, the estimated parameters do not converge to their
true values; this was dramatically shown in the paper by Astrom
and Wittenmark [1], also included in this volume. Early use of
Lyapunov theory to establish stability was restricted to mini
mum phase plants with relative degree one. Plants with a rela
tive degree higher than one posed a major problem [17] that was
not satisfactorily resolved until the late 1970s; indeed, the paper
by Goodwin, Ramadge, and Caines was a major contribution
to the resolution of this formidable problem and a substantial
stimulus to subsequent research.
Suppose the system to be controlled is described by
Ay
=Bu
where A and B are polynomials of degree nandm, respectively,
in the shift operator
q,
and
A
is monic; for continuous-time sys
tems q is replaced by the derivative operator
d
/
d t.
The system
may be re-parameterized (as shown in [1] and [15], the latter in
'state-space language') to obtain
Ey = Ry + Su
where E = E
1E
2
is monic and Hurwitz, and
£1
and E
2
have
degrees nand m, respectively. Hence (ignoring exponentially
decaying terms)
where
YE
:= (1 /
E)y, UE :=
(1 /E)u,
()
is a vector of the coeffi
cients of the polynomials Rand S, and the regressorvector ¢ has
components q i
y E,
i
= 0,
.. .
,
n
- 1, and q i
U
E,
i
= 0,
.. .
,
n.
The estimation equation
y
=
¢T
e
may be used in many differ
ent ways to provide an estimate
e
of the unknown parameter
e.
Supposethe desired closed-loop transferfunction is H = (N / E)
so that the desired output is
y* =
Hr , and that b.;
=
Sn
=
1 is
known. To determine the control, the system equation may be
written in the form
E2Y = RYE
l
+ SUEl = 1/JTe
where 1/J:=E
2
¢
(the vector 1/1 has componentsqiYE
l
, i
=
0, .. . ,
n - 1, and
qiUEl ' i
= 0,
. . .
, n). The degree of
E
1
is n so that
1/JT emay be written as
1JT
e+ u. If eisknown (and the system is
minimum phase), the control u may be chosen (as u
=
E2Y*
1JT
()) to cause the output to satisfy
E2Y =
1JT
e
+ U
=
E2Y*
so that the tracking error
y - y*
decays to zero (and all signals
remain bounded). If () is unknown, a tempting strategy is to
employ the certainty equivalent control u =
E2Y* -
1JTewith
the result that E
2
y* = 1/JT e and the output is now given by
E2Y
= E
2
y* + t/JTe
The tracking error er := Y - y* satisfies
e t =
e
+
er
where y:= ¢Teis the estimated output, e :=
Y -
Y
=
¢Teis
the prediction error (Monopoli's augmented error),
e
:=
() -
e
is the parameter error, and er :=
y -
y*) is an estimate of the
tracking error
eT.
Since
eT =
[(1/E2)l/ T]e -
(1/£2)[
l/IT OJ
it is also known as the 'swapping error'; eT is zero if eis con
stant, but otherwise depends on the rate of change of e(in the
continuous-time case it is proportional to (d / dt)e).
Monopoli [15] proposed use of the certainty equivalent con
trollaw inconjuctionwith a simple gradientestimator
(d/ dt)O
=
¢e; however, as was later pointed out, convergence of the
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tracking error to zero was not established except for the case
of unity relative degree. The first proof of convergence (for
continuous-time systems) was achieved in [4] by modifying the
controller; nonlinear dampingwas added to ensure convergence
of eT to zero; this enabled convergence of
er
(and bounded
ness of all signals) to be proven. Because
of
the complexity
of
the controller, and of the accompanying analysis, the paper [4]
did not have the impact it deserved. The second and much sim
pler proof (for discrete-time, systems), presented in this paper
by Goodwin, Ramadge, and Caines, achieved convergence by
modifying the
estimator;
the rate of change of
8
was reduced
(thereby reducing eT)by the introduction of error normalization
(replacing
e
in the estimator algorithm
bye := e/[I, + 1c/J1
2
]1/2).
Error normalization was independently proposed in [3].
The relevant result is Lemma 3.1 of the paper by Goodwin,
Ramadge, and Caines, which essentially states:
(a) Suppose the
estimator
is such that the normalizedpredic
tion errore
=
e/[1
+
1c/J1
2
]1/2 lies in
f
2
•
(b) Suppose the
controller
is such that the regressorvector c/J
satisfies the growth condition
1c/J(t)1
2
c[1+ max{le(r)1
2
I rE
[0, t]}
for all t o.
Then:
(i)
Ic/J(t)
I is uniformlybounded, and
(ii)
e(t) 0 as t
00.
The result is simple to state and prove. It provideda simple means
for establishing convergence (and boundedness of all signals) for
a wide range of adaptive controllers and contributed to an explo
sive re-awakening of interest in adaptive control.
An
important
feature of the result is itsmodularity: condition (a) on the estima
tor can be established independently of (b), i.e., independently
of
the control law. The result was presented in an electrifying
seminar at Yale University (New Haven, Connecticut) in late
1978 and was rapidly extended to the deterministic continuous
time case in [16] and [21]. All three papers appeared in the same
issue of the IEEETransactionsonAutomaticControl; ironically,
the paper by Goodwin, Ramadge, and Caines was the only one
that did not appear as a regular paper. A parameter estimation
perspective of the continuous-time results was given in
[5],
pro
viding an analogous modular decomposition of the conditions
for convergence. Stochastic versions of the paper by Goodwin,
Ramadge, and Caines and of
[5]
appeared, respectively, in
[7]
and [6]. Research on convergence and stability has continued to
this day. For example, the earlier nonlinear damping approach
of [4] was extended in [11] to deal with 'true' output nonlin
earities for which certainty equivalent adaptive controllers are
inadequate,andbackstepping
[13]
was developedas a systematic
tool for designing adaptive controllers for linear and nonlinear
systems with high relative degree. Furthermore, switching was
introduced to enforce convergence when structural properties
(e.g., relative degree) are unknown
[18],
and modifications were
made in the basic algorithm to ensure robustness
[19], [9], [12].
Researchresults of this period were rapidly consolidated in texts
such as [8], [20], [14], [2], [22], and [10].
The paper by Goodwin, Ramadge and Caines is an important
milestone in the evolution of adaptive control. It contributed
much to the richness of a subject that has progressed far and that
now appears poised for further significant advances.
REFERENCES
[1] K.1.
ASTROM AND
B.
WITTENMARK,
On self tuning regulators,
Auto
matica,
9:185-199, 1973.
[2] K.1.
ASTROM AND
B.
WITTENMARK,AdaptiveControl,
Addison-Wesley
(Reading,MA), 1989.
[3] B.
EGARDT, StabilityofAdaptiveControllers,
Springer-Verlag(NewYork),
1979.
[4] A. FEUER AND S. MORSE, Adaptivecontrolof single input, single output
linear systems,
IEEE
Trans.
Automat.
Contr., AC-23(4):557-569, 1978.
[5]
G.C. GOODWIN AND D.
Q. MAYNE, A
parameterestimationperspective
.of continuous timeadaptivecontrol, Automatica,23:57-70,1987.
[6] G.C.GOODWINAND D.Q. MAYNE, Continuous-timestochasticmodelref
erenceadaptivecontrol,
IEEE Trans. Automat.Contr.,
AC-36(ll):
1254
1263, November1991.
[7]
G.C. GOODWIN,P.J. RAMADGE, AND P.
E. CAINES,
Discretetime stochas
tic adaptivecontrol, SIAMJ. Contr. Optimiz.,19(6):829-853,1981.
[8]
G.C. GOODWIN AND K.S.
SIN,AdaptiveFiltering,Predictionand
Control,
PrenticeHall (EnglewoodCliffs,NJ), 1984.
[9] P.A. IOANNOU AND P.KOKOTOVIC,
AdaptiveSystemswithreducedmodels,
Lecture Notes in Control and Information Sciences,
Vol.
47, Springer
Verlag(NewYork),1983.
[10] P. A. IOANNOU
AND
1. SUN,
Robust Adaptive Control,
Prentice Hall
(EnglewoodCliffs,NJ), 1989.
[11] I. KANELLAKOPOULOS, P. V.
KOKOTOVIC, AND
A. S. MORSE, Adap
tiveoutput-feedbackcontrol of systemswithoutput nonlinearities, IEEE
Trans.
Automat.Contr., AC-37(11):1666-1682, 1992.
[12] G. KREISELLMEIER AND B. D. O.
ANDERSON,
Robust model reference
adaptive control,
IEEE Trans. Automat. Contr., AC-31(2): 127-132,
February,1986.
[13]
M.
KRSTIC,
I. KANELLAKOPOULIS, AND P.V.KOKOTOVIC, Nonlinearand
AdaptiveControlDesign, John
Wiley, New
York,
1995.
[14] P.R. KUMAR AND P.P. VARAIYA,
StochasticSystems:Estimation, Identi
ficationandAdaptive
Control, PlenumPress (NewYork),1986.
[15] R. V.
MONOPOLI,
Model reference adaptive control with an augmented
error signal, IEEE
Trans.
Automat.Contr.,
AC-19:474-482, 1974.
[16] A. S. MORSE, Global stability of parameter-adaptivecontrol systems,
IEEE
Trans.
Automat.
Contr., AC-25(3):433-439, 1980.
[17] A.S. MORSE, Overcomingthe obstacleof high relativedegree,
Journal
of theSocietyfor InstrumentandControlEngineers,
34:629--636,1995.
[18] A.S. MORSE, D. Q. MAYNE,
AND
G.C.
GOODWIN,
ApplicationsofHys
teresis Switchingin ParameterAdaptiveControl, IEEE
Trans.
Automat.
Contr., AC-37(11):1343-1354, 1992.
[19]
S.M.
NAIK,
P.R.
KUMAR,
AND
B.E.
YDSTIE,
Robust Continuous-Time
AdaptiveControl by ParameterProjection,
IEEE
Trans.
Automat.
Contr.,
AC-37(2):182-197,1992.
[20] K. S.
NARENDRA, Adaptiveand LearningSystems-Theory andApplica
tions,PlenumPress (NewYork),
1986.
[21]
K. S. NARENDRA, Y.-H. LIN, AND L. S. VALAVANI, Stable adaptivecon
troller design, Part II: Proof of stability , IEEE
Trans.
Automat.
Contr.,
AC-25(3):440-448, 1980.
[22] S.
SASTRY AND
M.
BODSON,AdaptiveControl: Stability, Convergence and
Robustness,
PrenticeHall (EnglewoodCliff,NJ), 1989.
D.Q.M. & L.L.
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II. PaOBUDlI STATBMENT
In tbiI paper we shall be coacemed with theadaptive control
of
liDear
time-iavariant
fiDite-dimeDlioDal
systemI
haviq
the lollowina state
apace repreaentation:
X(I+
1)-Ax(I)+B,,(I):
x(O)-xo (2.1)
1(1)-Cx(t) (2.2)
wha'eX(I), u(t),
y(l)
are the 11X 1 state vector, rX I input vector,
and
m x 1 output vector, respectively.
A staDdard result is that the
system
(2.1),
(2.2)
can
also
be represented
il l matrix IraetioD, or ARMA, form u
m.eaD-Iquue output
is
bouaded
wheDevcr
the
samplemean-square of
the
noiIe
is
bouaded.
HOWC\W,
the
.-.u
questioD
of
stability
remaiDI
uaauwerecllor atoebutic adaptive aJaorithmL
The study of discrete-time
determiIlistic aJaorithms is of
indepeDdent
intereat
but also
provides
iDsiaht
mto
stabilityqucatioDl
in the stoehutic
cu e (IS].
Recent work by 100000U
a d
Monopoli (13)
bu
been
concerned with the
exteDlioD of the
results in
(2) to
the cliIcrete-time
cue. AI
for the continuous
cue, the aupICIlted
error method is
used.
In tbiI
paperwepreIeIlt
11ft' resu1tI
nIatecI to clilCJ'ete-time
iatic
adaptiw
coatroL
Our approKh diffen from previous
work
in
several ~
respects
altbouP
ca1aiD upectI
of our
approach
are
iaspirecl by
the workof Feuer aDd
MOlle 5].
The
uaIysia
preIeIltecl here does
Dot
rely
upon the
use
of
aupDeDted
erron or awdlWy iD.putI. Moreover, the alpithms havea
Vf IIY simple
structure
and ue applicable to lDultipltHDput
multiple-output
systems
with rath. pnera1
UlUlDptioDI.
11le paper praenta
a paeraI
metbod
of
auIyIiI for dilcrete-time
detenD.iDiltic adaptive control alpithmL
The
JDethod is iDUitrated
by
.tablilbiDa
slobal
converpDCe
lo r
thJeesimple alpithmL
Fo r clarity
of
prIMIltatioD,
we shall
first
treat a
simple
IiDIIo-iDput siqIe-output
alpitbm
in
detail.The reaults
wiD
then
be
exteaded
to 0
siqIo-in
put aiqIe-output aJaoritlulll
iDdudiDa
thole hued on recunive least
squarea. Piully, die exteDIioD to DlaltipJe..iDput
multiple-output
systems
will
be prIMIlteeL
SiDce the results in
this
paper were
prIMIltecl
a
Dumber of
other
authon [16]-[18] have presented related resultl lor diacrete-time de
termiDiltic adaptive control
alpitbms..
Short
Papers
Dllaet&-11IIIe
_
Adapdve
Coatrol
GRAHAM
c.
GOODWIN,
. . . . . . .
JIIIB,
PETER. J.
RAMADGI , AND
PETER.
E.
CAINES. MJ I88I. IBI B
.4....
n i l Ur •••
c-..
..
.............. ' 6 I I c l
........
•••• rde
-..,--.It II
tIIIt wII-..
...... . ....
,
..
............................
I. IN'raoOUCl10N
A _
problem
il l
control theory hasbeen the questioD
01
the . . . . . . oIlimp1e.llot.lIy
COD t
adaptivecontrol
alpithmL
By t1IiIwe III8Ul aJaoritlulll which,
for
all
iDitial
I)'IteID aad aJaoritbm
C&1IIe
the outputl
of
a liveD IiDear
system
to _
track
a
daind
output IICIueDCet
aDd
achieve
this with
a bounded-iDput
IeqUeDCL
11lere is
a
CODIidcrable
amoUDt of literature
OD
continuous-time
determiDiltic adaptivecontrol alaoritbma[IJ.
However, it
is
0DIy
recendy
that aJobal stability aacl conv-aeace of these aJaoritlulll bas
been
studied UDcler a-aeraI
uaumptioDl.
Much
interest wa
paerated by the
iDDovative
CODfipration
pI'OpC)Ied. by
Moaopoli (2) whereby the feed
back
piDI were
directly
.timated and aD
aupleDted
error sipal aDd
awdlWy
iilput aipaII
were
introduced
to
avoid the use of
pure
dif·
fereDtiaton
il l
the aJaoritbm. UDfortunate1y,
as
pointed out
in [3J
the
8I'JUIIleD.tI
livea
il l
(2)
CODCeI DiDa
stability
are
incomplete. New proofs
for related
aJaorithma
have recently appeared
(4], (S]. In
(4]Narendra
and.
VaiavaDi
treat
the
cu e
where
the difference
in
orders between the
DUIUI'&tor
aad deDomiDator 01 the
system
traDller function (relative
etearee)
is laa thaD 01' equal
to
two. In (5], Feuer and Morse
propose
a
solution for
paeralliDear
systems without coDltrainta
OD
the relative
dep'ee. The alpithma il l [5] use the
auplented
error concept and
auxiliuy iDputi
U in [2J. The
Feuer
and
Morse result seems
to be
the
moat paeral
to
date for siDale-input siqle-output continuous-time
sya
tau. Howevw,
these
results are teclmicaJ1y involved and caDDot be
directly
applied to the dilcrete-time case.
11lere
hal
also been interelt in'dilcrete-time adaptive control for both
the determiDiatic ADd
Itocbutic
cue. This ana
baa particular
relevance
ill viewof the iDcreuiDa ut e of ctiaital tecJmolo&y in control applications
(6),(7).
ldUDI
[8],[9J
bU proposed a pneral techDique for
auaIyziDa
conver·
aence of diacrete-time Itocbutic adaptive alaoritbms.. However, in
this
auIyIiI a
q..aioa whichis ye t
to
be
reaoIved
ccmcems
the
boUDdednesa
of the . ... vuiabl-.
For
OM
particular
aJaorithm
[10], it
baa been
arpecI
iD [II] that the alpiduD poIIeII the property that the sample
(2.3)
with
appropriate initial conditioDS. In (2.3), A(q-I), BU(q-l)
( ;
1,··· ,m; j - l , · · · ,r ) denote scalar polynomials in the unit delay opera
tor q-l aDd the 'acton q- represent pure
time
delays.
Note
that it
is
not UIUIIled tbat the
system
(2.1),
(2.2)
is completely
controllable
or completely observable, DOl'
is
it
IIIUIDeCl
that (2.3) is
imclucible.
The systemwill be
required,
however,
to satisfy
the condi
tioDa of Lemma
3.2.
It is UIUIDed
that
the coefficientsin the
matrix.
A, B, C in (2.1),(2.2)
are UDbown and
that
the s ta te (t)
is
Dot
directly
m.surable.
A
feedback
coatrollaw
is
to be
dcaiped
to ltabilize
the system
and to
caue
the u t u ~
(y(I»,
to track
a liven reference
sequence
{y·(I)}.
Reprinted from
IEEE
Transactions on Automatic Control,
Vol. AC-25 , 1980, pp. 449-456.
487
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Specifically, we requireye'l
and u(t)
to be bounded uniformly in I, and
LentmtJ 3.2: For
1MsyJtem
(2.3)with r - m, and subjectto
lim y ~ ( I ) - Y I · { t ) - O
i - I , - · ·
,me
'
....
00
UI.
KEY Tl cHNlCAL
LBMMAS
(2.4)
[
zdu-dIBII{Z)
det :
Z ~ I - ~ B I ( Z )
(3.6)
Our
aualysil of
discrete-timemultivariable
adaptive
control a1p'ithma
for
Izi <1
WMn
will be buecI OD the loIlowiDa
technical results.
UmmtI
3.1:
11
14- min
tlu
I <j< ,
i - I , · · ·,m,
(3.7)
ax 1 ) , t + 4 ) 1 - ~
O<t<T
1<1<111
if
'11M , . , . u in COfUItUttI
m,.
4
whicll
an of T witA0 <III ] <
00 ,
0
< 4< 00
.rudI
that
hoof:
The rcault
ilataDdard ad simply followsfrom the
fact that
(3.6)eaaunI that the systemhu • stable inverse. 0
IIIthe
remainder of the
paper thae
reaul.
willbe used to
prove
Blobal
CODV...... of a Dumber of
adaptive
control aJaorithma. Sections
IV-VII will be concemed with
adaptive
control of
siqle-input
output
ayatemI.
SectiODI
vm and
IX
will
extend
these results
to
the
mul1iple-iDput
m u l t i p ~ u t p u t
cue..
(3.1)
(3.2)
(3.4)
(3.3)
wIleN0<CI <
00 ,
0<C
2
< co,
i l follow8 tMt
lim 1(1)-0
'
....
00
O<b.(t)<X<ao
and
O<b
2
(1)<K < oo
for
aliI>
0 tIIIIl
2)
COIIIlltitIII
lIer(t)H <
C,
+
C
2
max I.r(
1')1
0<.,.<,
lim s(/)2
-0
-+00
b l I ) + ~ t ) c r / ) T c r )
w1Mre (b.(I», ~ I » ,
IIIttI
(aI(/)} t ire retIlXG1tIr tIIItl (CJ(t)} i.r II
1WIIJl-tJ«IfJr .....,.eei . .
Alb}«1
10
1)
f I I I i n I I ~ CtJIIdltlort
tIItIl {UCJ I)JI} i8
bouItdttd.
Proof: II (I(/)} ill
a
bounded
sequence,
thea by (3.3) (IIcr(1)1I) is
a
bouDdecIleqUeDCe. Thea by (3.2) and(3.1) it follows that
lim 1(/)-0.
1-+00
IV. SINGU-1NPtrr
SlNGu-otrrPur SYSTBMS
It is
well
known
that
for the siD&le-input single-output(SISO)case, the
system output of (2.1), (2.2) can
be
described by
(3.S)
ad
Hence,
(4.1)
whero (u(t)}.
(yet)}
denote
the input and output sequences, respeo
lively, mel A(q-I) . B(q-I) are polynomial functiODS of the UDit delay
operator q-l ,
..4(q-I)-I+G,9-
1
+
.. . +1I,.q-
B(q-I)-bo+b1q-I+
.. •
+b ,q- '; bo+O
d
repJaeJltl the
system
time delay. The initial conditions
of
(2.1) arc
replaced by
iDitial
values
of
y(t),
0> t > -11,
and u(
t).
-
d >t»
-
d
-
m
The
loIlowiq 'UI1IIDptioDl will
be made about the
system.
A ru1It tiM Set 4:
a) d is Imown.
b) AD
upper
bound
for
and m
is
known.
c)
B(z)
hal all zeros strictly outside the closed unit disk. (This
is
neceaary
to easure
that the
control objective can
be achieved
with a
bounded-input
sequence.)
We
note thaltby
successive substitution, (4.1)
can
be rewritten u
> x
1/2
+X
I
/
2
U
cr(
t,.)
fI
:>
18(1,,)1
gl/2+ KI / ,
C. +
C
2
1r(t,,)
1] uaiDa (3.3)
and
(3.5).
lim
lor(
',.)1- 00
1,.-+00
Now IIIUJIle
that
(.t(t)}
is unbounded.
It follows
that
there
exists a
lubiequeDce (I,,)
such that
I.r< t)1
<
'8(,-)1 for
I <I .
Now aloDa
the
aublequenee
{t,,}
I
.r(t.) I> 18(1.)1 uaiDa
(3.2)
[
b. (
..>
+
1.)_(
t,.)T
G( . ] 1/'1 [ K
+
KII_(
',.)112]
1/2
lJ(t,,)1
where (y·(t)} is a reference sequence.
It
is assumed that (y·(t)} is
kIlowD
II
priori
and
that
lim
I .r(tJ I> _1_ >0
,--+ao
[ b . I , , + ~ I , , a t , , : r c r t l l
]1/2
K'/2('2
but this contradicta (3.1) IIld heDce the assumption that
(8(/)} is
un
bounded it falaead
the
remit
follows.
0
ID
order
to 1110 this
lemmaill
proviDa pobal CODveqence of
adaptive
control
alpithma
it
will be necessary
to
verify
(3.1)
(with .r(t)
interpre.
ted
u
the trackiDa error) aDd
to check
that IlllUlDPtioDa (3.2)
and (3.3)
are satiafied.
The nat lemma
will
be
used to
verify
that
the linear boundedDCSI
conditioD
(33)
is
.ti Ified
by an important e. . . . of linear timo-invariant
ayatemI.
This c1uacomspoDda to
tho.
tiDear timo-invariaDt systemI for
which the control objective
(2.4)
CU l
be
achieved with a bounded-input
sequence aDd lor which the trackiDa error
ca D
be reduced
to
zero i f
the
systaD
parametenare known.
y(t+ d)-a(q-
J)Y(/)
+ /l(q-I)u(t)
where
As previously
stated, the control objective is to achieve
lim [Y(I)-
y·(t)J-O
' ....00
(4.2)
(4.3)
(4.4)
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V.. SISO
PROJECTION
ALGoRITHM I
Let .
be the vector of systemparameters (dimensionp - n
+
m
+d).
for all values of tp(1
- d)
provided
0 <0(/) <2..
This
is
satisfied by
definition
(5.8). Then,
since
111(/)11
2
is
a bounded nonincreuing function
it converges.Setting
(S.I)
(I). -
,,(1- d)T;(/- l)
(S.13)
lben
(4..2) can be
written
y(t+d)_cp(t)T,O
where
(5.2)
[and DOtin.that
tI(I)[
-2+11(1)
, , (I-d)T,(I-d)
]
[1+f)(t-d)7 cp(t-d»)
Now
doIiDe
the output trackiDI error as
,<1)7'
-(y(t),· ·
. ..,(/-11+1), (/),··
·
,11(/-
m - d+ 1».
(5.3)
is boUDded away from zero, with a(/) defmed u in (S.8») we conclude,
from
(5.12),
that
+ d)
-
y(l+ d) - ,·(1 + d)
•
cp(
I)
T
0 -y.( 1+d).
By chOOliDa
(II( t)}
to satisfy
cp(/)T O·y·(/+d)
(5.4)
(5.5)
and hence
(5.14)
it
is evideat that the tractiDI error is
identically
zero.However,
since
0 Now using (5.13)and (5.11)
it
follows
that
is UDbaowD, we replace
(5.5)
by the
foUowinl adaptive aJaoritbm:
(5.9)
(5.10)
(5.15)
e(l)
[ 1+cp(t - d)
T
«p(
1-
d) ]
111
4-1 - d)T
-
0(/- ;) y\/
i- I [I
+4fJ(t-d)T.,(t-d)]1/2
, , ( I -d- i )
e(I)- _cp(l-d)Ti(,-d).
(I)
0<I 0 / - i)tp(t- d)T fP(t -d- ;)
[I +
'<,- tl)TCP(t_d)]1/2
[1+. t -
d-i T. , t -d- i)]1/2
f(1-1)
I
Hence,
[1
+cp(t-d-i)T <t-
d -
;)]1
12
• (t - i) . (5.16)
[I
+cp(t-d-
i)Trp(t-d - ;)]1/2
Nowby the Cauchy-Schwarz inequality and the fact
that la(t)1
<2
. f (1-d{ < I -d- i ) f(l-i).
[I
+.ct-d-i)TqJ(t-d-i)]
Then uaina (5.4)and S.7) wehave that
d - I
rz
«(I) . -4p( t-d) I ( / -d ) - 11(1-
i)
I- I
· [1+tp(/-d-
i . c t -d- ;»)1/2
I
2 ( 1 -
i) I
< [I+cp(I-d- j)Tcp(I_d_ i»)1/2 •
Then using (5.14) it follows that
7',-
lim tit)
1(/)
-0.
I--.CO
[1+9(t)T4p(t)]l/2
Proof: UIiDa the
dermitiOD
of i(t),
S.6)
may be rewritten as
i(t)
-
i ( / - l )
-
a(t)4p{t
- d)[l
+q>(t- d)T.(1
-
d)J-l
·,,(t-d)Ti(t-l). (S.lt)
i(t) - if1-
I)
+a(t).(1- d) [
1
+ .p(t - d) Tq> 1- d)]-1
T.
A
]
· [y(t)- cp(t-d) 1(/-1)
S.6)
tp(1)Ti(t)-Y·(I+d)
S.7)
where
i(t) is a
}I-vector
of reaIs depeadiDaon an
initial
vector i(O) and
ony(t'). 0<.,<1. u(1'), O<.,<t-dvia (5.6),
and
wherethe p in CODItaDt
a(/) iJ
computed u follows:
a(/)-1 if [(II +
I)th
component of right-hand side of
(5.6) (5.8)
evaluated
uaina
a(t)-I ]+O;
- y otherwise where y is a constant in the interval
(e:.2-e:). y. .1 and 0<<<< I.
Thischoice of pin constant prevents thecomputed coefficientof u(I)
in S.7) beiDa zwo.We
also
remark
that
the purpose 01 the
coeffICient
1
in the
term [I
+f(t -d)Tf ( / -d)]-1 of
S.6)
is
solely
to avoid diviaioD
by
zero.
ADy
positivec:oDltaDt could be used in place of the 1.
Apart
from
thellbove modificatioD,
the
al.orithm
(5.6)
is
an
orthogo
lUll projection
of1(/-1) onto
thehypersurfacey(/)-tp(t-d)T,.O.
In theauJyaiaof this alJorithm, we
wiD rust
show
that
the Euclidean
Dorm
of thevector i(,)- 1< - '0 is a
noniDcreainl
fUDCtion alODI the
trajectories 01 the a1lOrithm. This leada to a characterization of the
limitin. behaviorof the alaorithm which will allow us to use Lemma3.1
to establishpoba1 coDveraence.
lAmmJI
5.1:
Along
1'-
801ution.r
of
(5.6), (5.7),
Hence,
;(1)11
2
- Ui(t-l)tf2.tJ(t)[-2+11(1) ,,(I-d)TCP(I-d) ]
[1+.,( t - d)T
t - d) ]
.
i (1-I)Tf( I-d)f I -d)Ti( , - I )
<0
(5.12)
[I
+.p(/-d)T,,(t-d)]
, , ( t -d- ; )
[1
+,,(
t
- d - i)
T
q>( t - d_ ;)]1/2
. (1 - i) 1-
0
for
;-1,2, .. ' td-1.
(S.l7)
[I +
, ,(1-
d -
;)Ttp(t_
d - i ) t
/2
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but
TberefOl'et l I I iq (5.3)
lIt'<t-d)1I
<p{
m,+[max(l,mJ)
max
IY(1 )I}
1<..
<1
hotI: Lemma
5.1
CDIlII'eI that condition
(3.1)
of Lemma
3.1
is
aatiafiecl,
with 1(/) - e(/). the traekiq error,
and
0(1)-
cp(
1-
d)
thevector
defiD.ed
by
(5.3).
Also
6.(/)-1,
and 6
2
( / ) - I .
It
foBows that the UDiform
bouDdedaell
CODdition (3.2)
is
satisfied.
AasumptioD 4c) a d
Lemma3.2 ensure that
Ju(k-tI)f<m,+m.
max
IY(1')1 foralll
<k<t.
1
<or<t
(6.3)
However,
since
'0 is UDknown,
the
control law will be reeursively
estimated. The foUowina adaptive
aJaoritbm
will be considered:
i t - i t -d - ~ . , t - d [ l
+cp(t-
d)T <t-d»)-1 e(t) (6.4)
Po
( t)
_.,(/)Ti(/)
(6.S)
whereA, is a
flXecJ CODStaq,t
and
i(t) is
a p-vectorof reals
depending
on
d
initial
values 1(0),•• • , I (d- I ) and on y(.,), 0<., cr , u(T).
0<1 <
t
-
d
- 1
via
(6.4). Not e that (6.4) is
aetuaI1y d
separate recunioDl
interlaced. (It bas recently been pointed out (18] that it is also possible
to
analyzea _ p e recursion without
iaterlaciDa lII iq
a
different
technique
but the same aeneral priacipals.)
The aualysia of projection
a1aoritbm
n bas mudl in common with the
analysis
for
projection
alptbm
I.
We
wiD
therefore
merely state the
analop
of Lemma S.l
and
Theorem S.I for the algorithm (6.4),(6.S).
ummtJ 6.1:
DejiM
where
cp(t)T
- (
-y(t)_·· -y t -n+ 1), - u / - l · · ·
-u t -m-d+ l),y·(t+d»
,T _ (ft -
P
R' I )
o - . ••• ,«,,-It
I · · · ~ d - I
Po .
It
is evident that the
traekiD
error can be made
identicaUy zero
by
ehooaina
(u(t)l such that
(5.19)
im
[Y(/)-
y·(t)]-O.
t-.ao
Hence,
usiq (5.16),
(S.I7), and (5.14)
lim
e(l)
-0.
(S.18)
1-+00
[I
+e,(/-d)Tf {/-d)]1/2
Thiaeatabliahea
(S.IO).
0
Note that
we
do
Dot
prove, or claim, that i</)
converps
to 0
However,
the
weaker condition (S.lO) will
be
sufficient
to establilb
CODV eoce of the trackiDa
error
to
zero
and
boundedneu of
thesystem
iDputiand output&. These are the prime
properties
of concern in adap
tive control
77Ieorwm $.1:
Subj«1
to
AUIIIPf'liOlU 4a)-e);
if
tIIII
iIlgoritlun
(5.6),
(5.1) II f/IIIIIW to lite1Y.J/eIn (2.1).(2.2)( , . m-l) ,
tMn (y(/» tIItd
(u(t»
tIIW boIwI64 tw l
; / -
i( /) - 'o.
(6.6)
_
..
0
TIJsJ
IIlI(t+d)U
1-1I1 ( / ) l f
z
<
0 aJOIIg witla tlw
lOhdioru
of (6.4) and (6.5)
tmd
Hence,
l I , < t ~ d 1 I <JJ{m,+ [max(ltmJl
max
[Ie(r)l+mlJ}
1<.,<1
-C +C
2
max 1e(1')I; 0<C.<oo,O<C
2<oo
1<.,<1
and
it follows that the
linear
bounc1edness condition
(3.3)
is also satis
fied.
The
reault
DOW foDows by Lemma3.1 and by n o q
that boundedness
of {II
,(/)II)
eIIIURI
boundedDessof
{IY<
'>I}
and
{IN(t)I}·
0
IJo
0<
-: -
<2.
Ilo
(6.7)
o
We
Dote
that the condition 0<
flo/A,
<2
hu
been previously conjee.
tured (9), (10)
in
reprd to
stoehutic self-tunina
rep1aton uainlleast
squares. The condition
ca D
always be satisfied if the
sign
of IJo and an
upper bound for
the magnitude of Po
are
known.
Lemma 6.1
is
used to prove Theorem. 6.1 in the same lIl81U1er that
Lemma S.I is used to establish Theorem
S.1.
We obtain
the
foUowina
theorem in
this
way.
'1'heomn 6.1: Subject to AUIIIPf'tioIu 4a)-c) and for 0<Pol
<
2;
if
1M
tllgoritlun (6.4). (6.5) i.r app/i«J to
lite
qslma (1.1), (2.2), (yet)}
and
(u(t)} an bounded and
(7.1)
(6.6)
im
[y(t) - y·(t)J-O.
' ......00
VII. ADAPTIVE CoNn.OL
USING
R.Ect1RsJvE LBAST SQUARES
The
wide-spreaduse of recursive least squares
in
parameter estimation
indicatesthat it
may find
application
in
theadaptivecontrolcontext We
treat the unit
delay
cue 01algorithm I with the projection (5.6) replaced
by recursive least squares.
The adaptive control algorithm then becomes
; t - i t - l + a(I)p(t-2).(t-l)
[1+a(t)cp(/-l)Tp(t-2)cp(t-l)]
[ y( t) - cp t - 1)T
i(
t - 1) ]
e(/+ 4)-y(t+d) -
Y·(I+d)
-
Po(
lI(t)+crQy(t)·
••
~ _ I y t - I I
+
1)+
I
11
( t - l)
... +1I.:.+4-1 (t-m-d+
1)-
~ y . t + d »
-1Jo(
U(/)-
.,(1)7'0)
(6.2)
Let
VI. SISO PaOJBC1 lON
ALoOIU1llM
II
In this seetioo we present an algorithm differing from that of Section
V
in that
the cootrollaw
is
estimated dirccdy.This approach
is adopted
in [5), and essentially involves the factorization of fJo from (4.2). A
related
procedure
is used in the self-tuDing replator (10) where it
is
assumed
that the
valueof
IJo is mown.
AD advaDtqe
of
the aJaoritbm is that the precautions required in
Section
V to avoiddivilioo
by
zero in the calculation
of
the input are
DO
lonler neceaary.
However,
a disadvantap
is
that additionalinformation
is
required; specificaDy, we need to know the sign of fJo
and
an upper
bouDd lo r
ita
mapitude.
PactoriDl flo from (4.2)
yields
y(t+d)-JIo(crOy(t)+
.. .
+c(_ly(t-n+
1)+11(/)
+
lJ;u(t-I)··· +
fJ:,,+d_I (t-m-d+ 1». (6.1)
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(7.13)
P t -[ l - P(t-I),,(t)f(I)Ta(t+ 1) ]P(t-l) (7.2)
1+CP(t)TP(t-l)cp(t)a(t+1)
.(1)Ti(t>.y·(t+ I) (7.3)
where
p(t) is . Xp matrix and the recanion (1.2) is
8II1UIled
to be
iDitialized with p( -1) equal to any positivedefiDite matrix.
The ICI1ar G(t) in (7.1), (1.2) playa the samerole u in SectionV and
is
required 0D1y
to
avoid the noapDeric pOllibility of divisioD by zero in
(7.3)wh.
evaluatiq II{t).
Hence, G(I)-1 will almoIt always wort IDd
for tl(1)-1 we
observe
that
(1.1) aDd
(1.2) are the .tuldard
recursive
leut
IqlIU'eI
aJaorithm.
The sequence
{tI{t)}
may
be
choIeD
u
in
5.8).
u.. 1.1: AItmgwith 1M
8OIutItH&r of
(1.1), (1.2), (1.3)
.
/tIItctiolt
Y(1)-i(t)TP(t-l)-li(t) U
II bount/ed,
IIOIIMIt'tiN, PJOIIinc1W filfr jwIc
tioII
tIIIIl
Now
1,(t)1
2
le(t)1
2
[I+a(t)cp(t-l)T
p(t-2)cp(t-I)]
;>
[I+2U.(I-l)1I2<A.aIP(t-2)])]·
(7.12)
Hence,from (1.11) and
(7.12)
lim 1 <1)1
2
-0 .
''''00
[1
+2(Up(t-2»))U«p(t-l)U
2
]
TbiI
will
be
recopized
as
beiDa
condition
(3.1)
with .f(1)-
ce(t),
b.(I)
1, and
bi )
- 2(A..,J.p(t
- 2)D.
To establish the UDiform boundednesa condition (3.2) we proceed as
foUows. From (7.2) and the matrix inveniOD lemJDa,
r-
lim
.(1-1) ' ( I - I )
-0
'-.00
[1 +a(t)cp(I-l)7 P(t-2)CP(I-I)]1/2
i(t)-i(/)-,o-
Proof: Prom (7.1),
S.2),
• ...
a(I)P(I-2)tp(I- l )cp(t- l )T;(I- l )
1(1)-1(1-1)- .
[I +tI(t)CP(I-I)Tp(,-2).(t-I)]
Then uain
(1.2),
(7.4)
7.S)
pet) -1- P(I- l ) - I +a(t)cp(t)9(/)T.
Hence,
XTp(t)-l
x
>xTp(t - l )x
>A.ua[p(/- l)- l l
llx
Il
2
for each xEA'. (7.14)
Now
choose x as the eigenvector corresponding to the minimum
eigenvalue of
[P( t) - I
Then
from (7.14)
; ( I )_P(I- I )P(I-2)- I ; ( / - I ) .
Thus,
P t-l -I; I -P t-2 -Ii ,- l .
Now defllliD&
V(I) U i(,
-1)TP(t
-1);(1
- I ) wehave
A.m[P(I)-I] >A.m(P(t-l)
-I].
So AaJp(t)
-I]
is a nondecreasing function bounded below by
A-JP(- I ) - I ] -X -
I
>0.
(7.6) Hence from (7.13),
o<bz(l)
<2K. This establishes condition (3.2).
The
proofDOW
proceeds
U forTheorem
5.1.
0
u(t)
(8.1)
q
-tC-.B.....(q-l)
VIII. MULTIPU-1NPur
MULTD'LB-<>urPuT SYSTEMS
For the case m- r» 1, the system (2.1), (2.2) can be represented in the
form
where Ak(q - I)
and
BIc/ q - I) 1<k <m, I <1<;m are scalar polynomials
in the unit delay operator q - l with nonzero constant coefficient
Using
the m
identities l -A;(q-I )} j (q- l)+q-4Gt<q-l)where
(7.8)
• T
r-
lim 1(1-1)
.(1-1),(1-1) (1-1)
-0 .
,-.00 [I
+ tI(t)4p(t-l)T
p(t-2)cp(t-l)]
lim e(/) -0
, .....ao [1+a(t)cp(t-l)T
p
( I - 2)
cp
t - I»)' /2
Hence,
Y(t)-
Y(t-l)_;(t)T
p( l - l ) - I ; ( t )_ ; ( t - I )T
p
(t - 2) - Ii ( t - I ).
UI iq (1.6)
Y(t)- Y t - I -
[i I - i t - l ]T
P(t-2)-1i( t - l )
• •
_
-a(t)
i(t-I('PC
t - l7<t-I)Ti(t-l)
(7.1)
[1
+41(1)'<'-1)
p(t-2).,( I-
I)l
wherewe have
used (1.5). It is clear from (/.7) that V(I) is a ~
DODDeptive, DODiDcreuina
function and hence converp8.
Thus. from fl.1), aDd since
a(t) is
bounded away froJDzero,
where
and
(7.9)
4-
min
(dil}'
i-I,···,nI,
1<)< .
o (8.1) can be written
1'II«Jmn
1.1:
Subj«t to Amlnpliolu 4tI)-c) if I. aI,orilltm
(7.1), (1.2),
1.3) i8
. Ii. to 1M
IYstem (2.2) (r- m-l) ,
IMn {Yet)},
{fI(I)}
are
botI1ItJ«J
and
Proof.
From Lemma7.1
lim e(l) -0 .
(7.11)
''''00 [1 +a(t)cp(t-l)7 P(t-2)9(t-l»)1/1
lim [.,,(1)-)'-(1»)-0.
I-.ao
(7.10)
[
Y l t ~ d l
] _
[ a l { ~ - I ]y(t)
,.(1+
tJ...)
0 a .(q-l)
[
~ 1 I q - l ...
+ :
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f<a(t)<2-f
where O<e<1 and G(/)-l is no t an eipavalue of - R
- 1 ( ~ - l ) Y ( / )
with
where ~ ( t ) is -P,
( -
+
m(m,
vectorof realsdepending upon an
initial
vector
',(0) andy,(.,.), 0< <I,
u(T),
0<,. <1- d,via (9.4).
Oearly, it is critical
to
eusure that a solution to 9.S) exists for all t.
This
is guaranteed if the matrix of coefficients of u(t) in
(9.5)
is
noDliDplar this is
eDlured
by the procedure.
1) At
1-
0 the
arbitrary
initial value 1(0)
of
the
parameter atimate is
choeea. 10 that AuumptioD Set a is satiIfiod.
HO lce,
the aDaloaOUI
equatioDa to
(9.3)
evaluatedat
4 f ( 0 ) , y ( ~ ) ,
J< I < ..1(0)
He
solvablefor
11(0).
2) Fo r
I:>
1 the proceduJe of Lemma 9.1 below guarantees the 101va
bility01 thealgorithm equatiou for II(I).
Um1IIII
9.1:
I
ortIttr
tItat 1MIIttII1U t1/
cM/fidm/8
of
u(1)
in (9.5) i.r
1tOfLfbIIultv
for
Gill:> I U
u
III/fIdett
for
a(t) I (9.4) 10
be
cIIomt 4r
jollowl:
(9.6)
(9.7)
(9.8)
(9.9)
T ; - S, ,(t-I)
and
Y(/)
ro.,··· '011I]'
'. in (9.6)
is
the vectorof coefficientsof
II{t) in
,{ t
-
1), that is,
when
, in (9.6) is the vector of cbaDps in the coefficient of 11(/), that is,
where
and
, ( q - l ) - I i ( q - I ) B U ( q - l ) q 4 4 - ~ . 8.3)
It
can be seeD that
(8.2)
consists of a set of multiple-input siDIle-out
pu t
(MISO) systems haviDa a common input vector.
The
foUowiq
asaumptioDl willbemade about the system.
A .tltHt Set B:
a) d
1
, - · · ,tl.
are known.
b)
An
upper
bound
for the order
of
each
polynomial
in
(8.2)
is
knoWD.
c) The system(8.1) satisfies condition
det[
,lIIu-.t'Bu(z) .. • zd••-tllB. ,(Z)] ....o
rr: for Izi <1.
z4a-4l(.B..,(z)
•. •
4. ....B....(z)
Condition Ie) daervea commenL
Firat, for ally output colDpOnentYI' 1<i <m, there exists at least one
polynomial
9 - ~ B u < q - l ) .
I<j<m, for which the power of , -
associated with ita (DODZefO)
leadinl
coefficient is 4-
For
each such
po1yDomia1 theuaociated input appean in
y,(
t with the least
possible delay4.Evaluated at z-O condition Ie )
requires
the matrix of
these1eadinl coeIfic:ienti to be
1lODIiDp.1ar.
S e c o D ~ the aenericity of condition Be)(for the model (8.1») depends
upon
the
ini tial parameter ization of the system from
which
(8.1)
is
computed. TbiI cIepeadence is currently under investigation.
The control objective, u before,
is
to achieve
lim
[y,(/)-
yt(t»)-O
i - l , · · · ,m
t .....ao
IX. MIMOADAPrIVB CoNTllOL
where (/ ) is a reference sequence. It
is
assumed that each (yt( t )}
is 0,-
Sir C P I ( t - ~ ) ( l +9,(/-
~ ) T c p I ( t -
~ » - I
known
a
priori and that
ly,·(/)f <ml
< co for
aU
I, 1,- _.
· ( , , ( / )-
, ; ( t - ~ ) T I ( 1 -1»)]. (9.10)
(9.11)
1
<i
<em.im 1.1;(
t)
- y/.(1)1-0;
' ....00
Proof: Usins
Lemma (5.1) for each
i,
we have
. e.(t)
lim
f
-0
1/2
•
' ....00
[1 +
41>;(1
-
cp;(1 ]
Then:
i) R(O)is nODlinpiar by the initial choice of ; ~ O ) , ; - I,
. . .
t
m.
ii)
Assume R( I - l )
is
nODSinplar. Then from (9.11), usinJ a(t)+Ot
detR(t)-[detR(t-l)][det(l+a(t)R(t-I)-1
V(t»]
-ldetR(/-l»)(Q(/» [ ~ / » ) I + R / - I - I V(t)]
- 0 if and
ODly
if at/)
is
an eigenvalue of
-R( t - l ) - I
yet).
But the defini tion of a(/) euures a(t)-I is not an ei&envalue
of
- R(t-l)-IV(t),
hence
R(/)
is nonsingular
.. However, by i) R O)
is
nODSingular
and it foDows by induction R(t),
1
;>0 is noDSiDplar. 0
We note that the above choice of a(t) hu been included for technical
completeness
and that
a(
I) -
I
will
almost
always
work
since it is
a
Dongeneric occurrence for t to be an eigenvalue of -
R(/-l)-IY(/).
Also since - R(
1 - 1)- I
Y( t) bas only a fmite Dumber
of eigenvalues
it
is
always possible to find an a( t) to satisfy the lemma by computation of
the eigenvalues of R(1- 1)- I Y(I).
I Morem 9.1.·
Subject to Ammption.r &I)-c) if tlte algorithm
(9.4),(9.5)
if
applied10 tlw system
(2.1), (2.2)
with
r=m,
then
(y(/)}
and (II(I)} an
bou1ukd
and
(9.1)
(9.2)
(9.3)
<i<m.
· 1 + ~ ) - Y I t + ~ ) - , t t + 4 )
_ ~ ( t ) T ~ _ Y : ( / + ~ ) .
i , ( t ) - ~ ( t - l ) +a(t). , (t-d,)[ 1+ C P i ( t ~ t 4 ) T , , ( t - d,)]-I
T )
· ( Y I ( I ) - C P i ( t - ~ )
; ( t - I ) (9.4)
c p , ( t ) T ~ ( t ) _ , , { / + 1
«t <m (9.5)
492
Defme
where
It
is
evident
that
the
trackiD
error
may
be
made
identically
zero
if it
is
possible to choose
the
vector
u(/)
to satisfy
Let
'&
be the vector of parameters in (1,(9- . ) an d
fJll(q-I).
• • fJ,.(q-I).
Then (8.2)
may be written il l the
form
,(1+ cp,(t)T, . 1
<;
<m
Proof: The proof
will
be by induction and we first observe that
This section
wiD
be concerned with the multivariable versionsof
the
from (9.4)
and
(9.6)-(9.10)
adaptive control algorithms introduced in Sections V and VI. The
multivariable
version
of the allOritbmof SectionVII
also foDows analo- R( I) -
R(
1- I) +
a(
I) V( I).
gousty.
A.
MIMO
Proj«tioft
Algorithm I
Obviously (9.3) is a set of simultaneous equations in ..(t). Now the
matrix multiplying 11(/) is
nonsinplar
since in (8.3) det(diaIJi(z»-t at
z-O and Assumption
Ie ) holds. ~ e n c e
a
unique
solution
(1)
of (9.3)
exists at the true parameter value IJ.
Consider the
foUowilll
adaptive algorithm:
8/20/2019 Discrete-Time Multivariable Adaptive Control
http://slidepdf.com/reader/full/discrete-time-multivariable-adaptive-control 9/10
lim II,(/
..
)U-
co
I..
The proof
DOW
followa
that of Theorem. 5.1,except in the case that the
vector
)'(1) is unboundecL In
this case there
exists
a subsequence
(I,,)
such
that
and
IY/(
I + < IYJ('-)('- + forsome
I <al(
t..)<m
,aacl for all; I <I
<m and
tor
alii
<t,..
It
thea follOWl
by Lemma3.2 that
there
exist
coDltanti O<C
1
<co
IDd
O<C
2
<
oo
lUCIa
that
11.,('..)
< C I + C 2 I Y J , - t , . + ~ , . » l t
I <I<m.
SiDce
m is
fiDifet
there ail . . a further subsequence
{t,,}
of
the
sub
MqueDC8
{III} .uch
that
11., ,.,)11
<C.+C
2
Iy,(t
.
+4)1
forat
least
one
i, 1
<i <m
&lid (1,(,.,+4)} is ubouDded. The remaiDder of the proof then followl
dlat
01
Tbeonm
5.1wherewenote
that
where
d-max(d
1t
• • .Jt(J. i(t)T
is
aD ntXn
matrix of reala
depeactiq
OD II
iDitial
matrices I(f)T, 1
<; <
d aDd put data from
the
ayatem. P
isa
matrixof
COIIItaDII specified
aprlorl. AI in Section
VI.
<9.14)
repraeats
d interlaced recunioDa.
0DcI the recuraioDI (9.14), (9.1S) are iDitialized in a JDIImet thatIeadI
to • aique
IOIutioD f«
u(0) it illUlficieDt
to
e.uure that aD IUCCaliw
recunioJIIlead
to equatioallOlvable for
u(t),
I> I. In ..... to Lemma
9.1
we
have
foDowig. ..
U1nmtI 9.2: /hfttte I(t+tl)T _l(t+d)T- 'OT tIIId I«
K -
pro-
q
gT+K - KTg i.rporitiw
dI/iItit-,
t , tIlotw 1M tTtIjectorla
of (9.14),
(9. /5):
a)
trace[
i(I+d)Ti(t+d)]
-trace[
i(t)Ti(t)]
<0.
b)
lim e , t + ~
-0 , I<i
<me
t....
[I+,<I)Tcp(I)]I/2
Proof:
a) We caD
rewrite
(9.13)
UIiD
(9.1S) u
o
e,(I)I-IY,(
t) -
yt(
1)1·
Heace, from (9.14)
B. NINO Pro}«tioIJ Al,orltllm I I
From (8.2).
(8.3) the
fact
that
1'j(z)-1
for
z-o.
i - I , ·
. . , lit uad by
AuumptioD
Ie)
wecan
factor
out
the nonsiDgular
matrix
r
o
(-(IJ,(O)D
and
givinl
o
«t
<m.
X.
NON'LINBAIl
SYSTBMS
lim IY,(
t)
-
y;e(
t)I-O,
1-+00
or
trace(i(1+tI)Ti(t+ d)
-trace(
i(I)Ti(t»
_-trace[(KT+K-KTK f(t)T,(t)
)
(I+.<t)Tt(I)]
.(
i(t)T.,(t)[1+.,(t)T.,(t)r l.ct)Ti(t»
]
<
0 if gT+K
- KTK is positive dcfiDite.
b)
ill
the
proof
of Lemma (7.1)
it
foDows that
tim traee(x
r
+K-XTK
f(t)T,(t) )
1-+00
[I +.<t)I''<I)]
. ( i(1)TCP(t)[1 + cp(t)Ttp(t)]-l.,(t)Ti(t»_O.
SiDce
gT+
K - KTX is positive
defmite
then
tim
roll . ][1+9<t)r.,<t)J-
1
( t l ( l+
d) ·
.. e.(t+tJ...)]rol-o.
''''00 ·
-..(t+t.(.J
ThiaiJDpliea that (9.17) holds. 0
UIiDa
Lemma9.2and foDowiDa the proof of Theorem 9.1we havethe
folJowiDl.
1 II«nm 9.2:
Subject to
AulurptioJu
&I)-c), and K
T
+
K
- KTKpo.r;
ti w
definite, if
Q/pritltm (9.14). (9.15) ; appli«l to
1M.rystem
(2.1),
2.2) willir- m, ,hDJ
1_
fJ«10I .J y( I)
awl
II(
t)
tW bormd«IlIIId
Altho. the
aaalysia
in the
paper
baa been
carried
out for determinis
tic
Jinear
systems, it is clear that
it
could
be readily exteDded
to
certaiD
cIuIea of
nODlinear systems
of bowD form. The essential points are the
form 01 (5.2) or (6.2),
and the IiDear bound CODdiIiOD (3.3). The latter
point
would indicate that systems with cone bounded nODlinearitiea
(9.1S)
would satisfy
the
conditioD&.
493
Define
aDd
where
[
]_[JlI t;d
l
) ]_[JlW;d
l
) ]
e ,(t+
tI..)
y ,(t+tI..> y :(/+ t(..>
[
[
yf(t+dt) ]1
- r
o
u(t)+C(q-l)y(t)+D(q-l)u(t-l)-r;1 :
y':(t+d,..)
- r0<u( t) -
'OT.<
t»
(9.13)
where
'oT
ia
an
X,,'
matrix
whose
itb
row coataiDa the
parameten
from the ith fOWl of C(q-I). D(q-I), and E - r
o'.
f(/) is an
,,'X
1
vector
coataiDiDa
the appropriate delayed
veniou
of
y(t). u( t
and
y·(t):
4p 1)T
_ (
_ y(t)T, _
y t-
J)T,•. • , - fI(I_I)T,
-U(t-2)T, . . .
,yr(t+d.),···
,y,:(t+tJ..».
ADaIoaoUlly
to Section
VI we
introduce the followina adaptive control
alpithm,
[
el(t+d.) ]
i(t+d)T
_i( i)T-p : [l+ P(t)TfJ(t)]-I.(I)T
(9.14)
_.(1+
d .)
lIe
t) -
i(
If ,(t)
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xr, CoNCLUSION
The paper
hu
analyzed a
pneral class
of
discre.time
adaptive
control alaorithma and
has
abown that, under suitableconditions,
they
will
be
alobaI1y convcqent- The
alaorithma
have a
very
simple structure
and are applicable to both
sinl1e-input
siD&10-0utput and multiple-iDput
multiplo-output systemswith arbitrary time delaysprovidedonly that a
,table coatrol law exists to achieve zero trackiD error.
The
results
resolve a loDa staDdiDg question in adaptive control
reprdina
the
existence
of simple. atobally
converpnt
adaptive algorithms.
(1) I. D.
LaDdau.
A
lurvey
of model refa'OllCC adaptive tec1miqua-Tbeory aacl
appticaltall.
A
........
,
vo l to.
pp. 353-319,
1974-
(1) R.. V. MoaopoIi. Model
ref..-
adaptive
COIltrol
with al l a teeI error
Ii
lEU
TMu. AI.. . . , . e-tr., vol. ACI', pp. 474-415,
OcL
191...
(3) A. P ,
8.
R.
BarmiIb,
aDd A.
S. Mone. NAD astable dyDUDic:al
ayltem
UIOCiatecl
wi1b
aodeI nllNIICe
adaptive
control, IEBB
TPrIIIf.
A.....,.
Coftt,••
vo l AC23.
pp. 499-5001'
Ja e 1971.
(4J K. S. NaNDdra uc l L S. Valavui, -Slable adaptive
COIltroUer
cIeIipa-Direct
coatrol.·,BU
TNar. Alii....,.
CfIIIIr.. vaL
ACn.
pp. S70-S83.
A
....
1911.
(5)
A. 'eur ad S. Mane,
..
Adapti¥c QOIltrol
of
. . . . . . . . . . .
t tiMar
. , . . . . . .
IEEB
n- t . A...... , .
C . .
.•
wi .
ACD,
pp.
557-570. A .... 1971.
(fi) G. A. DuIoDt u d P. L Bitaqer, --seu-hIIliq coatrol
oI.litaaiUDl dioxicte
kiIa.
IBEE
n.u. Aw
..... CMIr., 'VOl. A ~ Z
pp.
532-531, A -. 1971.
(7) K. J. Aatrim,
U.
IIoriIIoa,
L
liUDIt
aDdB. WitteDmUk,
-rbeory
ud
appticatiou
of
. .
t
replaton. A........, 19. pp. 457-476.
1977.
(I) L
l i
AuIyI i I 01 recuniw .toebutic
aJaoritJu.,
IEEE TIwu.
A
......
C. ,.., voL
Aen .
pp. '51-575. A - . 1m .
(9) - , OD
poei, ,
na l traDII.
hact io .
aDd
tIM
COIlwrpl lCe
of IOID8
recursive
,.. 1BEE
TIwu.
,A.,..,.
C.. . , wL ACn,
pp.
539-551, 1m.
(10) K. J.
A.aimud
B. Wkteamark. -oa leIf-tuaiAa replatan, vol. 9.
lIP-
195-199, 1973.
1'1) L lsillDl
ad
B.
WiUlllUDal'k. OD •
ltabiIiziDa
property
01 adaptive
replaton,
. , . , . I I 'AC
sy .
__ rtIMtl/blltJlt.
TbiIiIi.
u.s.s.R.,
1976.
(11] 8.
I pftb. -A
uaiIied approIdl to
aaodeI
rei.....adaptiYe
. , . . . . .
aIlcllelf
tuaiAa
replatan, Dep. Automat. CoDtr., LuaclIDat. TedmoL Tech. Rep., Dec. 1m .
[131
T.t
.........L V.
MoDopoIi,
DiIcrete
IDOcIeI
refereace adaptive
CODtrol
with aD
........
error1ipaI, AIIItIfIIIIIktJ.
vo l 13.
lIP-
507-517. Sept.
1m.
(t4)
J.
L
WiIkaI, 5MbIIUy,.... . ofD,,. . , .aI
S)..... New York: 1970.
(IS) G.
Co
000cIwiD. P. J. Ramadp.
uc l
P. Eo CaiDeI, MDiIcreCe time
.todautic
adaptive
ooatrol.
SIAM
J. c.u,. OptiMiz•• to be pubIiIbecL
(lfi) A. S.
Mone,
-cHoba1atability of
parameter
adaptive
OODtrol.,.
..... Yale UDiv.•
S A IS Rep.
1I02L Mar.
1979.
(17]
x.. S. Nueadra
aDd
V-H.
LiB.
-Stable
ctilcrete adaptive ccmtrol, Yale UDiv., S •
IS
Rep. 79011' Mar.
1979.
[II)
B.1 pftb.
-Stability of modcI refcreace adaptive and
IeJf
tUDiq rqulaton, Cep.
Automat.
Caatr
Of
Lad
last.
TecbAo1.,
TedL
Rep., Dec. 1978.
494